Statistical properties of an iterated arithmetic mapping
The ``Collatz''-problem (or ``\(3x+ 1\)''- or ``Hasse''- or ``Syracuse''- or ``Kakutani''-problem) is to prove that for every \(n\in \mathbb{N}\) there exists a \(k\) with \(T^{(k)}(n)= 1\), where the function \(T(n)\) takes odd numbers \(n\) to \((3n+ 1)/2\) and even numbers \(n\) to \(n/2\). This problem (and a similar trifurcation process) is studied from a probabilistic viewpoint, while the authors do not present any real new theoretical material. Through computer experiments it is shown that the occurring parities exhibit all properties of random sequences. It is a remarkable fact that even statistical physicists are interested in this elementary number theoretical problem.
scientific article; zbMATH DE number 843783
scientific article